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H-curve
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By markdow   |   Watch
Published: December 16, 2009
The longest boundary is a relative of a Hilbert curve, but it doesn't reach everywhere within some rectangular regions. Those regions are filled by similar copies of the same curve. Like the Hilbert curve, it is the fixed point of a 2-symbol 2x2 block replacement system.


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anonymous's avatar
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AnonZytose's avatar
AnonZytoseHobbyist Digital Artist
Behold the power of H!
Sesquicentennial's avatar
it looks like a piece of technology like a printed circuit board or integrated circuit design.

Great choice of colors.
markdow's avatar
I recently read that chip designers use similar space-filling shapes to transmit the single clock signal across a chip, so delays are precisely known and regularly distributed.
hotmetal53's avatar
Yes. It is called a clock tree. Altho visually it would not be as regular as this except perhaps at a fairly large scale. The whole thing is automatically generated by software. The reason the regularity breaks down at small scale is because the leaf nodes are determined by the specific circuitry which is generally not so regular.
--
Allen Brown [link]
If you believe everything you read, don't read. --- Chinese proverb
markdow's avatar
So I'm guessing that the bulk of any delay is due to lags in circuit elements, not just propogation time across wires.
hotmetal53's avatar
Not so much any more. That was true 20 years ago. But now wire propagation is a very big deal. The transistors have gotten very fast. And the wires have gotten very thin, and therefore high resistance.

But a clock tree is more than wires. A wire will travel for a distance and terminate at an inverter (in this context, an amplifier) which then drives several wires each of which travel a distance... rinse and repeat.
--
Allen Brown [link]
By trying we can easily endure adversity. Another man's, I mean. --- Mark Twain
markdow's avatar
Oh, neat! Now I'll have to read up on this. I'm wondering how transmission speed depends on resistance, and whether the capacitance of neighboring wires comes into the calculation, and if the inverter delay times are significant, and ...

I'm starting a new project building a tuning fork clock, phase locked to an LC circuit, about 2 m high with about a 1 sec. period -- human scale so it can be poked and prodded, sort of an interactive demonstration of the dynamics of crystal oscillators. Now I'll be dreaming of a human scale clock tree! Maybe the tree could be use (slow) transverse waves on heavier chains for transmission.
hotmetal53's avatar
Yes, neighboring wire capacitance has a big effect. And it's ugly to model because if that signal is transitioning it injects noise into the clock. And that can create failures which are devilishly difficult to track down.

When I started in the industry decades ago the traces (or wires) were much wider than they were tall. Over time the width of the lines shrunk but the metal thickness didn't much. Eventually they crossed over. (I don't know what year that was.) Even 5 years ago wires were much taller than they were wide. As a result neighboring line capacitance became predominate.

The total clock tree delay is not terribly important. Since the clock keeps running all the time you know when the edge will happen. It's like catching a subway train. It doesn't much matter to you how long the train takes to make its circuit so long as another train comes along soon.

What's very important is clock skew or matching. Hence this structure we have been talking about.

One second period! Wow, that's slow, for both the LC and the tuning fork. I want to see this.

A clock tree is a whole lot like dominoes. There is a scene in V for Vendetta where he triggers a huge domino pattern. After spreading out to a massive area, it converges back to a single point. And that point is timed so two dominoes land at the same time. It is exquisitely well done.

Of course in a clock tree you don't reconverge. But if you didn't do that you wouldn't be able to tell how well timed it all was. Well maybe.

If suppose if you had each of the leafs of the tree make a noise you would be able to tell how well synchronized they all were.
--
Allen Brown [link]
The early bird gets the worm, but the *second* mouse gets the cheese.
markdow's avatar
Thanks!

Yes, the tall steel fork on a floating base is first part, and I'll send a pic when it's assembled. I haven't planned the LC circuit details, but I may use it as an excuse to make a large Leydon jar.

Reconvergence for demo purposes would be cool -- "tickticktick, tocktocktock".
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Sesquicentennial's avatar
that's interesting. I can see why it would be. If everything has the same distance from the clock source, the delay to each destination would be the same and everything on the chip would be in sync. Without it, if the clock is fast enough could cause different parts of the chips to be 180 degrees out of phase or worse with each other. I think it's pretty clever.
markdow's avatar
Yes, that's right. Amazing that the timing lags are large enough to get out of phase. The signal must travel at a significant fraction of the speed of light, across a centimeter give or take.
anonymous's avatar
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